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**EUROPEAN UNIVERSITY INSTITUTE, FLORENCE**
ECONOMICS DEPARTMENT
**WP 3 3 0 **
**EUR**
N ^ Ur° %

### 1

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**EUI Working Paper ECO No. 2000/6**

**The Impact of Selling Information on Competition**

**Karl H. Schlag**
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All rights reserved.

No part of this paper may be reproduced in any form without permission of the author.

© 2000 Karl H. Schlag Printed in Italy in April 2000 European University Institute

Badia Fiesolana I - 50016 San Domenico (FI)

© The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

### The Impact of Selling Information on

### Competition*

### Karl H. Schlag

### Economics Department

### European University Institute

### March 2000

**Abstract**

We consider a homogenous good oligopoly with identical consumers who learn about prices either by (sequentially) visiting firms or by con sulting a price agency who sells information about which firm charges the lowest price. In the sequential equilibrium with maximal trade and minimal search, prices are dispersed and consumers randomize between consulting a price agency and buying a t the first firm encountered. Low competition among price agencies induces maximal price dispersion. High com petition among firms leads to offers th at are either rip-offs or bargains where most consumers visit a price agency and firm profits are small. Keywords: price agency, intermediary, sequential search, price dispersion. JEL Classification: D43, D83, L ll, L13, L15.

‘We wish to thank Klaus Adam, Monika Schnitzer and Avner Shaked for helpful comments. The idea for this paper is based on a master’s thesis at the University of Bonn by Andreas Meis- ter. Financial support from the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303 at the University of Bonn is gratefully acknowledged.

© The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

© The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

**1 **

**In tr o d u ctio n**

As the number of products increases and markets globalize we find an increasing
demand for information, in particular, about where products are sold at what
price. This has been creating an increasing supply of information. The internet
is a growing source of free information; many sites offer information on the
location of the firm selling a given product cheapest (e.g., search routines among
internet book stores are very popular). W ith the expansion of the internet,
opportunity costs for finding the appropriate sites have been rising too. Sources
that explicitly sell information include newspapers, shopping clubs and price
*agencies. A price agency is a firm which sells information about the cheapest *
seller of a given product. In recent years, a particular form of price agency has
been rapidly spreading from Germany, Austria and Switzerland to the rest of
Europe. The consumer is charged a percentage of his savings either relative to
the lowest price known to him prior to his visit to the price agency or relative
to the producers’ recommended price.1

Consumers can profit two-fold from buying information. Directly, they are guided to the locations of firms who charge lower prices. Indirectly, the information given to some consumers increases competition between the firms to the benefit of all consumers. The service of the price agency is a public good. In particular, when a known set of firms can only be distinguished according to the price they charge for a given good then all consumers wishing to purchase this good will not choose to visit a price agency. Market prices would otherwise reflect perfect information and thus eliminate any additional value of visiting a price agency. Our paper contains the first analysis (known to us) where the price of information is endogenous in such a competitive goods market. The related literature (including [Hiinchen and von Ungern-Sternberg, 1985]) is discussed in the conclusion.

We aim to explore the impact of costly information providers on price for mation, profits, dispersion of information and on consumer behavior. We focus on the public good effects of price agencies and assume that their only service is to provide information about prices. We consider a simpler payment scheme of the price agencies than the one described above and assume that a price agency informs its customer about the lowest market price for a fixed fee (or tariff).

*1 (unverbindliche Prcisemfehhing in German)*

This price agency tariff is treated in three different settings: (i) exogenous, (ii) set by a monopoly or by colluding price agencies and (iii) determined through competition among price agencies. At the end of the paper we relax the assump tion that price agencies act honestly and discuss an alternative flexible tariff that provides the right incentives.

The underlying goods market is modelled in the simplest possible way. A
homogeneous good is supplied by finitely many identical firms without cost or
capacity constraints. Finitely many consumers each demand one unit of the
good. A finite number of price agencies each charge a fiat fee (or tariff) for
informing a consumer about which firm is charging the lowest price. In the
first part of the paper we consider the strategic interaction between consumers
and firms while keeping the tariff of price agencies fixed at an exogenously given
commonly known level. The sequence of moves is as follows. First firms simulta
neously set prices observable by price agencies but unobservable by consumers.
Then each consumer independently and sequentially gathers information about
prices, either directly by visiting one of the firms or indirectly by consulting a
price agency. Each time a consumer visits a firm he can decide to buy the good
at this firm, to visit some other firm, to visit a price agency or to terminate
search and to not buy the good at all. It is assumed costly for a consumer to
*gather information; the first visit to any firm costs s > *0.2

We choose to solve our model using standard game theory, ruling out non credible threats by focussing on sequential equilibria. In particular, deviations from equilibrium play can only be learned from observations. Thus we depart from much of the search literature (e.g., [Diamond, 1971], [Hanchen and von Ungern-Sternberg, 1985]) that assumes the actual distribution of prices to be known. In addition, we limit our attention to efficient (more precisely, welfare maximizing) equilibria where each consumer buys the good with the minimal number of visits to a firm. In these equilibria, consumers either go directly to a price agency or buy the good at the first firm they visit. Equilibrium conditions ensure that consumers do not prefer to continue their search (or to go to a price agency) after they have visited the first firm.

Typical for such sequential search models (see [McMillan and Rothschild, 1994]), there is no trade if the price agencies’ tariff is too large. W ithout potential customers coming via a price agency, firms have an incentive to raise their price

technically this means sequential search, without replacement and with free recall.

**2**
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as they take advantage of the additional search costs involved should a consumer choose to visit another firm. Notice that this argument is based on the fact that consumers do not learn about a price increase until they actually visit the firm. Thus, if no consumer visits a price agency, firms charge the monopoly price. Consequently, consumers choose not to buy the good and not to visit any firm as they cannot cover search costs of the first visit. A simple corollary is that some consumers will visit the price agency with positive probability whenever trade occurs in equilibrium.

When the services of price agencies are less expensive and some consumers visit a price agency, a firm which raises its price trades off earning more from uninformed consumers and being less likely to be the cheapest in the market, thus lowering the expected number of sales to consumers who visited a price agency. For a sufficiently small price agency tariff efficient equilibria exist and are completely characterized. In these equilibria, firms randomize among prices belonging to a closed interval. Expected market prices lie above the price agen cies’ tariff as consumers are indifferent between buying directly at the first firm and going to a price agency. When only few (two or three) firms are competing in the market, comparative statics reveal that higher search costs cause expected prices to decrease, which can make consumers better off. Although an increase in search costs directly reduces consumer utility, it also generates a substitution effect through making price agencies more attractive, with higher demand for price agencies’ services pushing down expected prices. When there are many firms, then most consumers go to a price agency and aggregate firm profits are very small. Firms then mostly either charge a very low (bargain) or a relatively high (rip-off) price (as in [Salop and Stiglitz, 1977]).

Next we include price agencies as players and assume th at they choose their (publicly observable) tariff before firms set prices. Colluding price agencies (such as a price agency monopoly) that maximize their joint profits act as if they do not care about profits and only choose their tariff to maximize price dispersion in the market. When individual search costs are small, then unlike the exogenous tariff setting analyzed above, consumers are always better off under lower search costs. When there are many firms then price agencies obtain almost all surplus.

Finally we briefly consider the effects of competition among price agencies. Should this lead to low tariffs then most consumers will visit the price agency and market prices will be low.

We proceed as follows. In Section 2 we present and analyze the basic

model without price agencies. In Section 3 we add price agencies and assume tariffs to be exogenous in Subsection 3.1. In Subsections 3.2 and 3.3 we then analyze collusion and competition. Section 4 provides the necessary adaptations to ensure that price agencies act truthfully. In Section 5 we discuss the related literature and conclude. The appendix contains the explicit calculations for the special case where there are either two or three firms in the market.

**2 **

**In d iv id u a l Search**

First we analyze market behavior without price agencies which also results when
price agencies’ tariffs are too high. Consider the following oligopoly market with
search frictions, n identical, risk neutral firms produce a homogeneous good at
marginal cost 0 (there are neither fix costs nor supply constraints). A finite
number of identical, risk neutral consumers each has demand for one unit of
the good and a reservation price of 1. To simplify notation, we normalize total
demand to 1 which makes all equations appear as if there is only one consumer.
Consumers only learn the price offered by a given firm after visiting this firm.
*We assume that a consumer incurs a cost s € (0,1) each time she visits a firm *
*for the first time, s is meant to reflect frictions when gathering information from *
an unfamiliar source. Later visits of the same firm are thus assumed costless.

We consider the following sequence of moves. At the outset, firms simul taneously choose their price for the good. These prices are not observed by the consumers. Then each consumer either chooses a firm to visit or decides not to buy the good. Once visiting a firm, a consumer can either purchase the good at this firm at the price offered, visit another, possibly previously visited firm, or decide to quit the search and to not buy the good from any firm. The game ends either when the good is purchased or when the consumer decides to terminate his search and to not buy the good at all.

*The strategy of a firm i can be identified with a cumulative distribution *
*of prices Fi. The strategy of a consumer can be very complicated, although two *
simple pure strategies are immediate: “do not search and hence do not purchase”
*and “purchase at any price at firm i" . Throughout this paper we will analyze *
*symmetric Nash equilibria in which each firm chooses the same c.d.f. F and each *
consumer uses the same purchasing strategy. Our game has many symmetric
Nash equilibria, e.g., each firm charges the price 0 and each consumer visits a

random firm and purchases the good there if this firm charges 0 and otherwise
does not purchase the good. Notice that consumers’ threat in this equilibrium
*not to buy when a firm charges price s/2 is not credible. Given search costs s, a *
consumer would be better off buying at s/2 than not purchasing the good at all
or purchasing the good at a different firm. In the following we restrict attention
to Nash equilibria that are not based on such incredible threats. Formally we
will restrict attention to sequential equilibria. Typical to models with sequential
search ([Diamond, 1971], [McMillan and Rothschild, 1994]), we obtain market
failure in equilibrium.

P ro p o sitio n 1* There is a unique sequential equilibrium: firms charge the mon*

*opoly price *1* and consumers immediately decide not to purchase the good.*

The proof of the inexistence of pure strategy equilibria can be stated ver
bally. Consumers are willing to pay any price below 1 when they are visiting a
*firm (since search costs s are sunk). Thus, all firms charge 1; the firm charging *
the minimum market price strictly below 1 can otherwise raise its price slightly
*(by less than s) without loosing its clients. Consumers anticipate these high *
prices and decide not to visit any firm (and thus not to purchase the good) as
*they cannot cover their search cost s > 0 of visiting the first firm. The more *
general proof for mixed strategies is analogous and therefore omitted.

**3 **

**Search in a M arket w ith P r ic e A g en cie s**

In models of sequential search there is too little competition among firms. Firms
exploit consumers visiting their firm which induces consumers not to start to
search in the first place. Things change when we add price agencies. A price
agency is assumed to know all prices in the market and sells this information to
*consumers at a publicly known price (or tariff) b > 0. Consumers are not able *
(or allowed) to resell their information to other consumers. When a consumer
acquires the services of the price agency, all information necessary for the trans
action are included. So we assume that the consumer does not incur any further
cost when going to the firm specified by the price agency. On the other hand,
the consumer’s visit to the price agency also involves individual costs which we
*assume for simplicity to be equal to s.*

**3.1 **

**E xogen ou s Tariffs**

*Assume at first that all price agencies charge the same exogenous price b > 0 *
for their services. Notice that the existence of price agencies does not preclude
market failure. It is immediate from the analysis in Section 2 that regardless of
*the size of b there is always a sequential equilibrium in which all firms charge *
the monopoly price 1 and all consumers decide at the outset to not purchase the
good. In order for there to be trade in a sequential equilibrium, some consumers
must visit the price agency with positive probability. A visit to a price agency is
only profitable when there is uncertainty about the prices in the market. Since
consumers know the equilibrium strategy of each firm this means that some
firms choose a mixed pricing strategy. However, not all consumers will choose
to visit the price agency. Otherwise, Bertrand competition is induced among
the firms as only firms offering the lowest price will get customers. Firms will
consequently offer price 0 and consumers will purchase directly at firms without
going to the price agency first.

In the rest of the paper we restrict attention to symmetric sequential equi
libria that are efficient in the sense that each consumer buys the good and the
*expected number of times the cost s is incurred is minimized. In these equilib*
ria, each consumer randomizes between going directly to a firm to purchase the
good and going directly to a price agency to then purchase the good at a firm
charging the lowest price.

In the following we will derive necessary properties of these equilibria. Let
7 6 (0,1) be the probability that a consumer goes directly to a price agency.
*Equilibrium prices will be contained in [0,1]. A firm raising her price from p *
*to p' yields a trade-off between lowering the probability of receiving customers *
*that first went to the price agency (provided F (p) < F (p')) and raising profits *
*per sale. Hence, firms will randomize over a closed interval of prices [a,0] , i.e., *

*F is strictly increasing on (a,/?) with [*q*, 0} C [0,1]. Let p = f pdF (p) be the *
expected price offered in equilibrium by a firm.

*Next we calculate F as a function of 0- In order to give incentives to firms *
to choose the same pricing strategy, each consumer who decides to go directly to
a firm must be equally likely to visit each firm. Hence, with probability £ (1 — 7)
*a consumer visits firm i and purchases the good there. Firm i also sells to all *
consumers that first went to the price agency if it chose the lowest price in the

*market. Indifference of firm i over prices p € [a,/?] implies*
i ( l - 7 ) + 7( l - F ( p ) )n—1

*Since F (a) = *0 we obtain from (1) that
1 “ 7
1 — 7 + rry
• P = - ( ! - 7 ) •/? • _{n}

**0**

**0**

**(i)**so

*F{p) = F (p,7,/3) =*if 0 < p < 1-7

_{7+n7^}

*^ T ^ P < P < 0*(2) 1 if

where the corresponding density is given by

*0 * 1

*f i.P) = n — *1

*0 < p < l*

*0 — p \*

ri7 / *( 0 ~ p ) p*

*for a < p < 0. In particular, the minimal price a always lies strictly above 0. *
*Given p = J p f (p) dp, we obtain*
p =
*0*
*n — *1

*0*

/ l - 7 ^
I " ' l i m /
V n 7 *)*1" 1 lim / V « 7

*)*1 * - >

*J-.*

*T ^ 0 0 ~ P*

*0*

*~ P \ "*

*P*

*~ P*

*dp*

*P*

*dp*

### (3)

*using change of variables p = p//3 so p = 0p and dp = 0dp. Hence, h (*7*) := p /0 *
*is independent of 0.*

Each consumer buys the good and is indifferent between going directly to a firm and going first to a price agency. Hence,

*s + b + pBn = s + p < *1

_{(4)}

*where pBk denotes the expected minimum price in the market when k firms *
*randomize independently according to F. Moreover, a consumer who decides to *
*go directly to a firm will not purchase the good if the price is above s + p since *
*he always has the option to visit a second firm. Hence, 0 < s + p.*

*In the following we will derive additional properties of F when p = s + p *
*as we shall show below that P = s + p holds in these equilibria. Notice that this *
means that any consumer who goes directly to a firm where the good is offered
*at the maximal price p is indifferent between buying the good at this firm and *
*visiting another firm. Since p = s + p = s + p h (*7) we obtain

*The profit of a firm equals what it earns by charging the highest price p = s + p. *
It must also earn an equal share of what consumers pay directly at firms and
what they pay for the good after visiting the price agency. Hence,

*(s + p) *(1 - 7) - = (1 - 7*) - p + *7*- p Bn , *

*n * *n * *n*

which holds if and only if

*P B n* - I s .
Thus,

*b = p —* 1* \ a =* 1 - 1 1* s.*
1* — h * 7

(5)

This leads us to the first main result.

*P ro p o sitio n 2 An efficient symmetric sequential equilibrium exists if and only*

*if there exists *7 6 (0,1*) that satisfies*

*s * *s * *( * 1 l \

*P - s + p = --- - *< 1* and b = p - pB„ = P ---- = --- ---s . * (6)

1 *— h * 7 \1* — h * 7/

*In such an equilibrium, each firm chooses prices according to F satisfying (2). *
*Each consumer goes directly to the price agency with probability *7* and goes *

*equally likely to one of the n firms with probability *1 — 7*. A consumer buys the *

*good at any firm she visits if this price is less or equal to s + p.*

*The expected price nc paid by a consumer, the expected profit of a firm *
7*Xf and the expected profit of the price agency irp in a given efficient symmetric *

sequential equilibrium are given by

*nc*
*nf*
7Tp
*p = P - s =*
1* - h "*
1* — 'y*
*- n (' - l } P = M T C*I f '
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*P ro o f. All we are left to show is 0 = s + p is necessary and that *7 6 (0,1)
satisfying (6) actually induces a symmetric efficient sequential equilibrium. No
*tice that a firm has no incentive to offer a price below a since a firm setting a *
already almost surely attracts all consumers who visit a price agency.

First we show that a consumer will never choose to go to the price agency
*after visiting k firms. Let x be the smallest observed price. Assume th at x = 0. *
Then the expenditure including search cost of going to the price agency equals

*s + b+p B(n-k)- Since s + *6*+ps(n-*:) > s + b+pBn = s + p > 0 a consumer offered *
*the good at price 0 strictly prefers buying over going to the price agency. Now *
*consider the case where x < 0. Then the disutility of going to the price agency *
equals

*s *

*+ b + J *

*p*

*■*

*(n — k) f*

**(p) [1 —**

*F*

**(p)]"-*-1 **

*dp*

**+ [1 —**

*F*

**(a:)]"-* **

*x*

*.*

*Using the fact that the derivative of the above expression with respect to x *
equals

(1* ~ F ( x ) r k*

*we obtain that the consumer strictly prefers buying at a firm who charges x over *
going to the price agency.

Finally, we show that each consumer buys at the first firm visited. Consider
*a consumer’s behavior after searching n — *1* firms with x being the lowest price *
offered by these firms. Expenditure including search cost of going to the n-th
firm equals

*s + [ P f (*p*) dp + (1 - F (x)) x ,*

*Jo*

expenditure when not going to the n-th firm is x. We will show that the indi vidual never strictly prefers to visit the n-th firm, i.e., that

*x < s + [ p f (p) dp + (1 — F (x)) x * (7)

*Jo*

*given 0 < s + p . Notice that (7) is true when x = 0. Moreover, the derivative of *
*the right hand side with respect to x for x < 0 equals *1* — F (x) < lan d hence *
(7) holds for all x € [a,/?].

Now consider the choice of whether to visit the (n — l)-th firm. Since the
n-th firm will never be visited it is as if there is not an n-th firm. By induction
on the number of firms it then follows that each consumer buys at the first firm
*visited as long as 0 < s + p. In particular, this implies th at a firm will choose*

*0 = s + p since sales at price 0 axe only made'to consumers who do not consult *

*the price agency and consumers will buy if p < s + p. This completes the proof.*

Next we discuss when such equilibria exist together with some of their
properties. It is easily verified that 6 (0) = 1, 6 (1*) = 0 and that h is continuous *
and decreasing in 7 as

*h' ( 7) =* * _{( n - 1)7(1 -}*1

_{7}

_{)}

*-h +*

_{1}1 - 7

— 7 + 717 *< *0

*Given 0 < s < 1 let *7 = 7* (s) € (0,1) be the unique solution to h (*7*) = 1 — s. *
Then it follows from (6) that the probability that a consumer visits a price
agency in one of our equilibria is at least 7. Now consider some 7 6 [7,1). Then
*the price distribution F is well defined which implies that p — pBn > *0. So if

then (6) is satisfied. Hence, any 7 € [7,1) (and only these) can be supported
by an efficient sequential equilibrium if the tariff 6 is chosen appropriately. In
particular, 7 = 7 and 6= 1*— s*/7 > 0 can be supported in which case the
largest price offered equals the monopoly price 1* (i.e., 0 = *1*), expected price p *
is maximal and consumers are indifferent between not buying and buying the
good (either at a random firm or via the price agency). When 7 > 7 then all
prices are bounded away from 1 and consumers strictly prefer to purchase the
good. For 7 close to 1, the tariff 6 supporting this equilibrium as well as the
expected price is close to 0* (while the maximal price 0 is close to s) as p > pgn, *

*P — PBn is continuous in *7* and p gets small as *7 gets close to 1. In particular,

*C o ro llary 3 An efficient equilibrium exists when 0 < 6 < 1 — s /*7.

It remains unclear whether efficient equilibria only exist if 6 < 1 — s/7
and whether equilibrium demand for price agencies is decreasing in the tariff
(although both statements are easily verified for n € {2,3}, see appendix).
Of course, efficient equilibria (as well as any other sequential equilibria with
*trade) fail to exist if s is sufficiently small for given *6 since a consumer strictly
prefers visiting each firm over going to the price agency whenever (n — 1*) s < b. *
*Similarly, equilibria with trade fail to exist when s is sufficiently large for given *6

*since consumers will not participate in search whenever b > *1* — s as this implies *
*that p+ s > 1. When there are only three firms then the set of (s, b) constellations *
where efficient equilibria exist can be calculated explicitly (calculations given in
the appendix, regions illustrated in Figure 1).

*When price agencies are sufficiently attractive (i.e., their tariff b is suf*
*ficiently small) for given s then efficient equilibria exist. Demand *7 for their
services will be arbitrarily close to 1 and the expected market price will be arbi
trarily close to 0. Almost all surplus will go to the consumers. When there are
*either two or three firms, then comparative statics for intermediate values of b *
are also available as the demand for price agencies is decreasing in their price.
*Lowering b decreases the maximal, the minimal and the expected price of the *
good. This makes consumers better off and firms worse off (changes in profits
of the price agency will be analyzed in the next section).

Efficient equilibria do not exist for extreme values of the individual search
*cost s given b. Again, comparative statics can be obtained for intermediate *
*values only when n € {2,3} . Increasing s leads to two opposite effects. Since *

*p is increasing in s for given *7 we find that a higher individual search cost has
a tendency to raise prices. On the other hand, there is a substitution effect
*between individual search and going to the price agency. Increasing s is like *
*decreasing b. For given b, as s increases, the demand for services of the price *
*agencies increases. This causes a decrease in normalized prices h = p//3. In *
the appendix we show that the latter effect dominates the former as we find
*that expected prices are decreasing in the search costs s. Of course, expected *
*prices remain above b so that eventually, when s is too large then an efficient *
*equilibrium fails to exist as b = p — pBn can no longer be satisfied. Whether or *
*not consumers are in fact better off when search costs s increase depends on the *
*change of s + p. The substitution effect is strongest when search costs are small *
*(i.e., when equilibrium demand is small). For n — 2 we find that consumers *
are better off if and only if the equilibrium demand is below 0.635, for n = 3
this is the case when 7* is below 0.786. Figures 2 and 3 show the graph of p as *
*a function of s where b = 0.05 and n = 2 or n = 3. In particular, we find that *
*there is only a relatively small range of search costs s where an increase in s *
*makes the consumers better off (s € (0.17,0.274) for n = 2, s € (0.0 66,0.14) *
*for n = 3).*

The calculations of the equilibrium strategies when there axe either two or three firms (see appendix) also reveal the explicit pricing strategy of a firm.

*While the price density is decreasing for n = 2, it is u-shaped for n = 3; there is *
a higher concentration of prices at the extreme ends of the price interval [a, /?]
*with the density at /? being unbounded (the later remains true for n > 3).*

Equilibrium characteristics can also be derived when the market is very
competitive. In particular, our next proposition reveals that efficient equilibria
*exist when n is sufficiently large for given b and s with b < 1 — s. Price agencies *
become very attractive and each consumer consults one with an arbitrarily high
probability. Firms either compete for consumers who visit the price agency and
charge a price close to 0 (bargain) or they exploit the uninformed and charge a
high price (rip-off) slightly above the price agencies’ tariff (actually a price close
*to b -f s). Asymptotically, bargains and rip-offs are offered with probabilities *

*s / (b + s) and b/ (b + s) and the expected market price equals b. All surplus is *

divided between price agencies and consumers.

*P ro p o s itio n 4 For b < 1 — s and *£ 6* (0, b + s) there exists no such that n > no *

*implies there exists *7n G (1 — e, 1*) with*

*K [in) < *1
*b =* 1 - — Is
7n /
*max {|Fn (e) - s / (b + s )| ,\Fn (b + s - e)*
*~ K (7„)*
*s / ( b + s )|} < £*

P ro o f. Following Corollary 3 the proof of the existence of an efficient equi
*librium for sufficiently large n and given b is complete once we show that *
lim info.,,» 7*n = 1 where hn (qn) = 1 — s. Assume that liminf*7n = 1* — 6 < 1. *
Setting 7 = 7*n and 0 n = s / { l - h (*7J ) = 1,

*lim sup [1 — Fn (p)] = lim sup* = 1

for any given p < 1*. Thus, hn (*7n) > 1 — s holds once n is sufficiently large.
*This however contradicts the fact that 1 = s + hn (*7n) holds for all n. Hence,
lim inf 7n = 1.

*Now consider the equilibria induced by tariff b and *7 = 7*n. Since j n > *7n,
we obtain limn_oo7*n = 1. Hence, (3 = b + s/-yn tends to b + s as n —> *00. (5)
*implies that psn tends to *0* and hence lim ,,^^ p (n) = l i m ^ ^ b + pBn = b.*

Let

*v := *1 — lim inf_{n—}_{* oo}

*For any fixed p * 6 (0, 6* + s ), we obtain lim inf,,-,,» Fn (p) = * ia Hence,
lim inf„_oo p (n) *= (1 — i/)(b + s). Comparing this to the fact that*
limn_ 00* p(n) = b we obtain i' = s*/(6* + s). Repeating the same argument for *
lim sup then completes the proof. ■

**3.2 **

**M o n o p o listic P rice A gencies and C ollusion**

While the price agencies’ tariff was assumed exogenous above, we now include
price agencies as players. For simplicity, assume zero cost for price agencies to
obtain information about prices charged. The results below are readily gener
alized to include fix and constant marginal costs. At first we consider a single
*monopolistic price agency who publicly determines its tariff b before prices are *
set and consumers search. This price agency aims to maximize its expected
profits 7Tp = 7* • b.*

*In this enlarged game, each possible choice b of the price agency induces *
a subgame. In each of these subgames there are multiple sequential equilibria
including one that induces market failure (see the beginning of Section 3.1). In
the following, we focus on efficient sequential equilibria in which trade is efficient
*in each subgame in which this is possible; when b is chosen such that there exists *
7 € [7,1] with

then firms and consumers follow an efficient sequential equilibrium of this sub
*game as characterized in Proposition 2. Since b is chosen to maximize expected *
profits, an alternative interpretation of the analysis of this section is that we are
searching for the efficient sequential equilibrium that maximizes the joint profits
attainable by colluding price agencies.

*In our next result we measure price dispersion by the normalized differ*
ence between average price charged by firms and the minimal price charged in
*the market; prices p are normalized by dividing by 0. So expected price dis*
*persion in an efficient equilibrium equals (p — PBn) /0- Our next results shows *
that maximum profits are obtained precisely when expected price dispersion is
maximal.
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*P ro p o sitio n 5 Consider the equilibria characterized in Proposition 2. Then*

*i. The profit np of a price agency is maximal if and only if the expected price *
*dispersion {p — pBn) / P Is maximal.*

*ii. There exist s°, *7** € (0,1) such that:*

*(a) I f s < s° then the profit maximum is attained when demand equals *

7**; in this case all consumers strictly prefer to purchase the good and *

*a, P, p, 7Tf, b and *7*Tp are proportional to s.*

*(b) I f s > s° then profit maximizing demand is greater than *7*. *If in *
*addition n € {2,3} then consumers are indifferent between buying *
*and not buying and expected prices are decreasing in s.*

P ro o f. The proof of part 1 is complete once we show arg max < -—777—r - l|/i (7) < 1 - s

*7€(0,1) [1 - h( ~f )*

*= arg max {h (*7*) - g (*7*) |h (*7) < 1 - s}
7€(0,1)

*where g — Pbu/ P ■ Following (5), g = g (*7) = (1 — 7*) (1 — h) /*7 and hence,

*h — g = *1 — (1* — h) */7 is maximized if and only if (1 — /z) /7 minimized if and
only if 7*/ (1 — h) is maximized. Thus the claim is proven.*

*Notice that h (*7*) < 1 — s if and only if P (*7*) = s / (1 — h (*7)) < 1. Let 7*
*be the largest maximizer of the heterogeneity h — g, i.e.,*

7* = max j arg max *{h *(7) *— g *(7) : 0 < 7 < 1} j .

Then 7* **€ ***(0,1). Let s° = ***1 — ***h (*7**). Then s < s° implies h (*7*) **< **1 **— ***s and *

hence,

7* £ arg max {76 (7*) \h (*7) < 1 — s} .

*The fact that a , 3, p, ttf, b and *7*Tp £ire all proportional to s follows from *

(6) and the fact that 7** does not depend on s.*

*If s > s° then h (*7**) > 1 — s so h (*7*) < 1 — s implies *7 > 7*.

*For n € (2,3} , b is decreasing in *7 and price heterogeneity is concave in
7. *Consequently, s > s° implies th at demand equals *7 and 6 = 1 — s/7 in the
*profit maximizing efficient equilibrium. In this case, p = 1 • h (*7*) = 1 — s. ■*

Table 8 gives the explicit values of the equilibrium parameters from Propo
*sition 5 when n € {2,3} and s < s° (calculations contained in the appendix). *
*Notice how the comparative statics in search cost s have changed in comparison *
*to our analysis in Section 3.1 for a fixed tariff b. When s is small, the price *
agency chooses a small tariff which makes consumers strictly prefer to buy the
good. Increasing s, the price agency offsets increased demand for its services due
*to increase in individual search cost s by increasing its tariff b. Demand *7 in fact
remains unchanged and hence expected prices increase as firms take advantage
*of higher search costs. When s is above s°, the price agency charges the highest *
tariff under which consumers are still willing to participate in the search for the
*good. Increases in s are now compensated by decreases in *6 as the price agency
has to make itself more attractive to compensate for higher individual search
costs. Expected price decreases to keep total consumer utility unchanged.

Next we consider the profit maximizing sequential equilibrium found in
Proposition 5 when there are many firms. As we already saw in the case where
tariffs were fixed (Proposition 4), price agencies become very attractive when
there are many firms. Thus, it is not astonishing that we find below that the
monopolistic price agency sets its tariff close to the maximal level 1* — s and *
still attracts most consumers. The rest of the results for fixed tariffs carry over.
Most prices are either bargains close to 0 or rip-offs which are now close to
the monopoly level 1. Approximately a fraction 1* — s of the firms offer a rip-off *
*resulting in an expected market price close to the maximal level 1 — s. Almost all *
consumers go to the price agency, they are nearly indifferent to not participating
*in the search. In particular, this means that the cutoff level s° from Proposition *
*5 tends to 0 as n tends to infinity.*

*P ro p o sitio n 6 For given s > 0, consider a sequence n = 1,2,.. of efficient *

*sequential equilibria indexed by the number of firms in the market where efficient *
*trade occurs in subgames whenever possible. Then the equilibrium parameters *
*satisfy*

lim q„ = 0, lim *= lim Q. = *1*, lim p„ = lim bn = *1* — s*

*n — oo * *n —*mx> n —*00 n —*oo n —» oo

*andlim^Fn (p,^n, 0 n) = s for any 0< p < *1.

P ro o f. Write equilibrium parameters as a function of the equilibrium value of
demand 7. Let 7’ be the profit maximizing level and let 7*n solve hn (*7*n) = 1 — s.*

Following the proof of Proposition 4, limn^ oc,7n = 1. Since

*P n ( i n )* - 7; - s > / ? „ (7n ) • 7„ - S = 7n ~ *S*

we o b ta in l i m * ^

*0n *

(7 ; ) = lim n_ 0O7 ; = 1 w hich im plies l i m ^ ^ a , , (7 ; ) = 0
a n d *limn-.oo bn*(7 *) = 1

*— s.*U sing th e fact th a t pn (7 n )

*= 8n (*7

**) — s*we also o b ta in limn-.ooPn (7 ; ) = 1 - s.

Finally, let

Then lim infn—00* Fn (p, *7*, /? (7*)) = 1*/ for 0 < p < 1 which implies lim infn_ QO *

*pn (*7*) = 1* — u. Since limn^ooPn (*7*) = 1* — s we obtain v = s. Completing this *
argument using lim sup instead completes the proof. ■

**3.3 **

**C o m p etin g P rice A gen cies**

Up to now we assumed that price agencies colluded, or alternatively, that there is
a single price agency who acts as a monopolist. Consider now at least two price
agencies competing against each other. Instead of explicitly specifying demand
and calculating equilibria we discuss the impact of competition with the help of
*comparative statics in the tariff b. Consider competition among price agencies *
that leads to all price agencies charging the same tariff where more competition
*leads to a lower value of the market tariff b. The discussion in Section 3.1 on *
page 11 reveals that demand for price agencies will be arbitrarily close to 1
and expected market prices arbitrarily close to 0 if competition among the price
agencies reduces tariffs close to 0. When either two or three firms are supplying
the good then any increase in competition leads to a higher demand for services
of the price agencies and lower expected market prices, making consumers better
off. However, increased competition need not result in lower price dispersion in
the goods market as we observed above th at price dispersion increases initially
when tariffs lie above the joint profit maximizing level.

1/ = 1 — lim inf © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

**4 **

**M oral H azard and V ariable Tariff S tru ctu res**

In our model, as in [Hanchen and von Ungern-Sternberg, 1985], we assume that price agencies actually inquire about the prices promised and then reveal the lowest price truthfully. However, as consumers cannot verify the information provided by the price agency, there is moral hazard. A price agency that incurs a positive cost with each price inquiry would choose to save costs of gathering information and would only check the price of one firm. Consequently, our equilibrium would brake down. One way out of this problem is to include public monitoring of the price agencies. Here we briefly discuss an alternative where we provide the right incentives to a price agency by considering a variable tariff.

It is plausible that a price agency as a firm with many employees has a technology that makes it profitable to sample prices simultaneously instead of in sequence. Consider a tariff system in which the payment to the price agency is decreasing in the price revealed to the consumer. Then, given a sufficiently large number of consumers, the price agency will choose to learn about all prices and to reveal the lowest price to its customers. Variable tariff systems can be found among price agencies acting in Germany where a customer without a price quote researched on his own pays a percentage of the savings relative to the manufacturer’s recommended price.3

To see how such a variable tariff affects our model, assume that the tariff of
the price agency equals a percentage of the savings relative to the largest possible
market price, i.e. it charges r (/? — pgn) where r € [0,1] is the parameter chosen
by the price agency. The indifference condition of consumers analogous to (4)
*becomes s + r (/? - pan) + PBn = s + p < *1 and hence

*= P - P B n _ * *1 + h - *1
*0 - P B n* **7 - ( 1 ---y) ( 1 ***- h ) '*

All results then extend directly to this alternative tariff structure, the only
*changes being that b has to be replaced by r in the statements. The values of r *
needed to complete table (8) are r2 = 0.155 and r.3 = 0.297.

3On the other hand, some price agencies charge a tariff that is increasing in the price they find. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

**5 **

**C o n clu d in g R em arks**

Information intermediaries such as price agencies facilitate trade and spread information among consumers. They profit from price dispersion and the degree of information in the market, parameters they influence when they fix the price for their services. To obtain a more detailed understanding of their impact, we set up a simple model where price agencies provide information about prices in a homogeneous good oligopoly. We consider sequential search so that consumers are always free to visit one more firm if they are not pleased with the prices previous encountered. Consumers remember which firms they have visited in the past and may go back to any of these at no cost (formally, sampling is without replacement and with free recall). The fact th at consumers remember which firms they have visited complicates the analysis substantially so we rest ambitions to find all sequential equilibria and focus on the efficient ones.4

We find price dispersion where the expected market price inducing trade is always greater than the price agency tariff. Maximal price dispersion obtains when competition among price agencies is weak or when there are many firms. Colluding price agencies deliberately set their tariffs to induce maximal price dispersion as this maximizes their profits. When there are many firms, price agencies become very attractive to consumers and most consumers visit a price agency. Firms consequently face fierce competition for business attracted via a price agency. This leads to extreme price dispersion as firms either offer bargains or rip-offs.

The equilibrium price dispersion uncovered in the seminal paper of Salop and Stiglitz [1977] is driven by heterogeneous consumers. Consumers with low search costs get the bargains while those with high search costs are left with a random draw among the bargains and the rip-offs. Only pure strategy equilibria are considered. Whether or not there is price dispersion depends on the relative search costs. In our model, consumers are identical, differences in their informa tion is endogenous. Given many firms we find an analogous result, the informed (which are the customers of the price agencies) get the bargains whereas the others get a random draw. Similar to Burdett and Judd [1983] and Shiloney [1977], we find that ex ante consumer heterogeneity is not necessary to obtain price dispersion. Price dispersion obtains in our model whenever there is trade

4 It is a simple excersize to verify that the equilibrium we characterize are the only sequential equilibrium in which trade occurs when consumers sample with replacement.

in a sequential equilibrium (see the beginning of Section 2), existence requires
that firms choose a mixed pricing strategy. The price distributions generalize
the findings of Gale [1988] who considers two firms and an exogenous proportion
of informed consumers. In particular, each firm randomizes among an infinite
number of different prices. Since we consider only a finite number of firms, our
*efficient equilibria thus cannot be reinterpreted in a setting in which firms do not *
randomize and consumers only know the distribution of prices in the market.

In many models (e.g., [Salop and Stiglitz, 1977], [Burdett and Judd, 1983],
[Hanchen and von Ungern-Sternberg, 1985]), consumers have a fixed sample size
and thus are not allowed to continue their search if they only observed high
prices. Thus, some firms charge the monopoly price whenever there is price
dispersion. In our model of sequential search, consumers are never forced to buy
the good. Consequently, in an efficient equilibrium, the maximal price charged
*by a firm never differs from expected price by more than the search cost s. In *
particular, when the price agency tariff is small, then the maximal market price
*is just above s.*

When there are only few firms (n = 2 or 3) then we can derive more spe
*cific results. Demand for price agency services is decreasing in the tariff b, in *
particular efficient equilibria are unique. Both the expected and the maximal
market price are decreasing in the tariff of the price agencies. However, the
opposite happens when individual search costs become small. We find a result
similar to Samuelson and Zhang [1992] that both expected and maximal price
*increase when the individual search cost s decreases. When s decreases, substi*
tution effects cause the services of the price agency to become relatively more
expensive. This leads to a decrease in the demand for price agencies, consumers
are less informed and competition among firms declines. Of course, when search
*costs s and the tariff b decline by the same percentage then it is immediate that *
prices decrease too (see (6)).

The only other model (we know) analyzing strategic considerations of price intermediaries when present in goods markets is due to Hanchen and von Ungern- Sternberg [1985]. While the basic questions analyzed in their paper are very similar to ours, their underlying model remains very different. In their “cir cular road” market, goods are extremely differentiated (where differentiation is not geographical); no two firms provide the same good for the same con sumer. Their intermediary not only provides information about the prices but also about the different characteristics of the goods being offered. Lack of com

petition among firms enables all firms to charge the same price in equilibrium while all consumers visit the price intermediary. In particular, in equilibrium, the information intermediary only provides information on which product is most preferred to a consumer. We set out to investigate some aspects surrounding price intermediation of the form provided by the price agencies found in Europe. These intermediaries provide information only about location and price (includ ing transportation cost to the consumer) of a given good. Here the setting of [Hanchen and von Ungern-Sternberg, 1985] seems less applicable. On the side, an interesting extension of either price intermediation model is to assume that only an unknown subset of the firms offer the desired good.

In the most popular tariff employed in Germany, price agencies charge a proportion (typically 30-33%) of the savings relative to a price the consumer has researched on his own. Notice th at our calculations cannot be used to rationalize the use of such tariffs as we focussed on equilibria where consumers who visit a price agency have no information about prices charged in the market. Moreover, as noted in Footnote 4, only these equilibria emerge under the common simpli fication that consumers forget which firms they have visited (sampling with replacement and without recall). Thus, it remains an intriguing topic for future research to analyze circumstances where these tariffs are used in equilibrium.

**R eferen ces**

*[1] K. Burdett and K.L. Judd (1983), Equilibrium price dispersion, Economet*

*rica 51, 955-970.*

*[2] P. Diamond (1971), A model of price adjustment, J. Econ. Theory 3, 156- *
168.

[3] D. Gale (1988), Price setting and competition in a simple duopoly model,

*Quart. J. Econ. 103, 729-739.*

[4] T. Hanchen and T. von Ungern-Sternberg (1985), Information costs, inter
*mediation and equilibrium price, Economica 52, 407-419.*

[5] J. McMillan and M. Rothschild (1994), Handbook of Game Theory, Voi. 2, R.J. Aumann, S. Hart (eds), Elsevier Science B.V., 906-927.

[6] S. Salop and J.E. Stiglitz (1977), Bargains and ripoffs: A model of monop
*olistic competition, Rev. Econ. Stud. 44, 493-510.*

*[7j L. Samuelson and J. Zhang (1992), Search costs and prices, Economics Let*

*ters 38, 55-60.*

[8*] Y. Shilony (1977), Mixed pricing in oligopoly, J. Econ. Theory 14, 373-388.*

**A **

**T w o or T h ree firms**

In the following we calculate the efficient sequential equilibrium characterized
in Proposition 5 when there are either two or three firms in the market. When
*there are only two firms (i.e., n = 2) then solving the integral in (3) we obtain*

*!»(?)
*f i ( p )*
*P2*
*02 h )*
*PB2* ( l )
1 _ 1 ^ - **7**^

**(02~P**

A
2
1 - 7
2 7p2
**(02~P**

*for ~ ^ 0 2 < P < 02*1 + 7 & M 7 ) =

_{27}

_{1}

_{ — }

_{7}______ _________ s 2 7 _ ( i _ 7 ) l n i±a

*= & - - =*27 27 — (1 — 7) ln i=?

*T*

*= 0292 (*7

*) = 0\*1 - 7

*02 ( 1 - 7*In 1 + 7 1- 7

*Here we find the equilibrium tariff h2 (*7) to be decreasing in the proportion
*of consumers going to the price agency (Figure 4 shows b2 (*7*) / s as a function *
of 7). It is easily verified that 762 (7) is convex in 7 and attains its maximum
0.1159s when 72 « 0.635 (Figure 5 shows 762 (7) / s together with the price
*dispersion ( p ~ P B2) / P as functions of *7*). Consequently, s° = 1 — h2 (72) ~ *
*0.569 and we obtain for s < s° the profit maximizing level *75 with 7*Tc = p2 ~ *
0.758s, 7*Tf « 0.321s, b2 = 0.183s and irP « 0.116s. For s > s°, the profit *

maximizing demand 72* solves h ( j 2) = 1 — s and hence *7 2 > 72- The density
and c.d.f. of the price distribution for s = 0.1 corresponding to 7 2 is shown in
Figure 6 ([a,/3] ~ [0.393s, 1.757s]).
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For the alternative linear tariff system presented in Section 5 the inverse
*demand r as a function of *7* for n = 2 is shown in Figure 7, in particular, *
r (7 ‘) = 0.155.

*Similar calculations for the case of three firms (i.e., n = 3) yields*

F (p ) =
/ (p) =
*P =*

**03 = **

**03 =**

**Ò**

### 3

** =**

, _ f o r *< P < h*V 37p 1 + 27

*03 / I ~ 7*1

**2**/?3 'Z 1 ~ 7

**2**37

*PÌ2\[0 3 ~ V*

**2 ^**

7T / 1 — 47
— - arcsin „ ■ ■ ■ —
2 V 1 + 27
2^ - \ / l - 7 [f - arcsin ( j ^ ) ]
*- 4 ( 1 - 7 ) v /Jy + y j l - ' r n - 2v /l - 7 arcsin*4 7 \/3 7

**- 7v/l **

- 77T + 27v /1 - 7arcsin ( y ^ )
Again, we find that 63 is decreasing in 7 and that 6 3 7 is concave. The profit of the price agency 6 3 7 attains is maximum 0.29s at 7 « 0.681. Thus we obtain for

*s < s° = *1 — /1.3 ( 7 3 ) « 0.528 that the profit maximizing level 7 3 « 0.681 which

yields p3 » 0.89s, 7T/ « 0.2s, 62 ~ 0.426s and 7Tp « 0.29s. The corresponding
density and c.d.f. for s = 0.1 are shown in Figure 8* ([a, (5\ = [0.256s, 1.894s]).*

Finally, we provide the main calculations used to show how p changes in s
*for given b. Given*

**“ M **

**(r^À - **

### 7

**) **

*-differentiating b = w(~) (s)) s with respect to s we obtain*

**0**

**7'(s)**

*w’ (7 ) •*7

*' (s) • s + w (*7)

*—w*--- 7 7 T

_{s • w‘ (}_{7}

_{)}> 0 and hence,

*h! (*7)

*w*1

*— h w‘ (*7)

*d s P*1 ^ (7) /i (1 - /i);7' (s)s = 1

*- h*© The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

*For n = 2 this implies*
*w' (*7)
*d*
7 , n l + 7
27 1 — 7
- In
1 + 7 1 - 7
= _ J L i n i ± 7 +
272 1 - 7 7(1+7)
+ _ 2
(2 7- ( 1- 7) l n | ^ ) *V *
1
9 s*P ■* _{2-y }_{I}_{-7}
/
**27**
7 j 1 + 7
*( _ *1 i_ 1±2 1 1 *) ( * 27______ l \ ^
\ 2 ^ * 1 - 7 ^ 7 ( 1 + 7 ) / y 2 7 —( 1 —7) I n 7 J
(l _ *ln l± r) I *2— ^7 ln^7__, +
l1 27 l n i - 7 J “ ( 2 7 - 0 - 7 ) , „ 4 ) v +(27- ( l -7) l n | ^ ) / 2*J*
*< 0*

where numerical calculations reveal that p > —1 holds if and only if 7 > a r g m a x { ( p - p Sll)//3} « 0.635.

For n = 3 we obtain

*h'( *7) ^x/7 ____1 __

872^3(1- 7) 27 ( l + 2 7 ) '

Given tc := 6/ s similar though more length calculations show that Jlp < 0 where numerical calculations reveal that Jjp > - 1 holds if and only if 7 > 0.786.

Table 8: Equilibrium parameters when there are either two or three firms
present.
*Tl * *S° * *b * 7* *p * *7Tj * *7Tp*
2 0.569 0.183s 0.635 0.758s 0.321s 0.116s (8)
3 0.528 0.426s 0.681 0.89s 0.2s 0.29s
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**b 4 **

1 .
**/ b=2s**

0.8
0.6
**' . b=1-s **

**0.4**

**0.2**

### /

**0 q**

**0 2 0.4 0.6 0.8**

**i *s**

Figure 1: Shaded area: Parameter range where an efficient sequential equilibr ium exists. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

1**j**
0.8
0.6
0.4
0.2
P
0 0 0.2 0.4 0.6 0.8 1 S

*Figure 2: Expected price p as a funtion of search cost s when n = 2 and b = 0.05.*

A
1 ■
0.8
0.6
| i
**0.4**
0.2
P
0 0 **0.2** **0.4** **0.6** 0.8 **{ *** s

*Figure 3: Expected price p as a funtion of search cost s when n = 3 and b = 0.05.*

▲
**0.4 :**

**b**

### s

**0.2**0.1

**0 0**

**0.2**

**0.4**

**0.6**

**0.8**

**1 *Y**

Figure 4: Aggregate indirect demand of price agencies per unit search cost when

*n = *2.

Figure 5: Aggregate price agency profits per individual search cost and price
*dispersion when n = *2.
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*A*
35 .
30 •
25 j
I
20
15

### j

10### i

5 ;**0**.

**price density f**

0.05 0.1 0.15 ' 0.2** P
Figure 6*: Equilibrium price density when s = 0.1 and n = 2.*

0.25
**0**.**2**’
0 .1 5
0.1 j
0.05
\

### \

0 0 0.2 0.4 0.6 0 .8 1 YFigure 7: Inverse demand of price agencies under the alternative tariff system.

4
25
20 | ** _{l}**
15 ;

**p r i c e d e n s i t y f**10 •

**\**

### \

5 . 0 0**0.05**

**0.1**

**0.15**

Figure 8*: Equilibrium price density when s =*

**0.2 ** **p**
*0.1 and n = 3.* ©
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